This paper introduces a novel approach for estimating the infinity norm of the inverse of monotone (inverse-positive) matrices and totally non-negative matrices. The proposed method is based on a partition of the index set into two complementary subsets. By considering partial row sums associated with this partition, the problem is reduced to the analysis of an auxiliary 2 × 2 matrix that captures the aggregated interaction between the subsets. The main result shows that if the constructed 2 × 2 matrix preserves the monotonicity property, then an explicit upper bound for the infinity norm of the inverse can be derived. This reduction principle enables greater flexibility in applications, as the partition can be adapted to the structure of the matrix, such as sparsity or block patterns. Furthermore, the new bounds generalize existing row-sum-based estimates, which can be viewed as the limit case corresponding to a trivial partition. The approach is extended to totally non-negative matrices via a standard sign transformation, preserving the structure of the obtained bounds. Numerical examples demonstrate that the proposed estimates can be significantly sharper than classical bounds, highlighting their practical relevance in stability analysis and numerical linear algebra.
This paper develops diagonalization-based algorithms for all-at-once linear systems arising from linear delay differential equations. The fully discrete all-at-once systems are approximated by their α -circulant counterparts and then decoupled in the frequency domain, providing the basis for the proposed solvers. First, we propose an α -circulant preconditioner for GMRES solvers. By employing a generalized Bendixson theorem, we derive a spectral enclosure for the preconditioned operator and characterize how the eigenvalue clustering around unity depends on α . Second, we develop a parallel low-rank correction method which decomposes the correction equation into a collection of independent rank-one generalized Sylvester-type matrix equations. These subproblems are solved by a two-sided Krylov subspace projection method. We analyze the low-rank structure of the subproblem solutions and characterize the polynomial nature of the underlying approximation spaces. The theoretical speedup models illustrate the algorithmic parallel potential of the two methods. Numerical experiments validate the robust convergence and computational effectiveness of the proposed methods.
This work presents a novel unfitted boundary algebraic equation method for solving three-dimensional elliptic partial differential equations in complex geometries. The developed method can be regarded as a finite difference version of the boundary integral method. We demonstrate that replacing finite auxiliary domains with free-space lattice Green’s functions streamlines the computation of difference potentials, enabling matrix-free implementations and significant reductions of computational cost. We establish theoretical foundations by proving the equivalence of densities obtained from finite and infinite auxiliary domains when the grids align, and by showing the equivalence between direct formulations in the difference potentials framework and indirect single- and double-layer formulations. Spectral analysis of the resulting boundary systems reveals that the discrete single-layer matrix has a condition number growing linearly with boundary size, while the discrete double-layer matrix remains uniformly bounded. The method is validated through matrix-free numerical experiments on Poisson’s and modified Helmholtz equations in 3D implicitly defined geometries, including ellipsoids, bumped spheres, and tori, demonstrating optimal second-order convergence rates and computational efficiency. Numerical results also reveal that the double-layer formulation requires fewer GMRES iterations for smooth geometries, compared to the single-layer formulations. This framework naturally extends to unbounded domains and provides a foundation for applications to more complex systems like Helmholtz and Stokes equations.
The discretizations of two- and three-dimensional spatial fractional diffusion equations with the shifted finite-difference formulas of the Grünwald-Letnikov type can result in discrete linear systems whose coefficient matrices are equal to the sum of the identity matrix and two diagonal-times-block-Toeplitz with Toeplitz-block matrices or block-Toeplitz with each block being block-Toeplitz with Toeplitz-block matrices. For these discrete spatial fractional diffusion matrices, we construct two block fast dominant Hermitian splitting preconditioners based on circulant and τ -matrix approximations to further accelerate the convergence rates of Krylov subspace methods. Theoretical analyses demonstrate that most of the eigenvalues of the corresponding preconditioned matrices are clustered in a complex disk centered at 1 with a radius less than 1. In addition, the numerical results show that the constructed preconditioners can effectively solve the discrete linear systems of higher-dimensional spatial fractional diffusion equations.
We study a numerical scheme for a hydrodynamic model of smectic-A liquid crystals, which incorporates velocity–pressure variables (u,p) and the order parameter of smectic-A. The model is a highly nonlinear system that couples the incompressible Navier–Stokes equations with a fourth-order parabolic equation for the layer variable ϕ , which is endowed with periodic boundary conditions. We first introduce a first-order decoupled scalar auxiliary variable (SAV) scheme for numerically solving the smectic-A liquid crystal model and present an efficient implementation. Moreover, we prove that the scheme is uniquely solvable and energy stable. Furthermore, we carry out an error estimate for the scheme. Various numerical experiments are presented to verify our theoretical results and simulate the dynamics of the system.
Gegenbauer–Sobolev polynomials are an important class of Sobolev orthogonal polynomials. We prove a novel recurrence relation for this sequence of polynomials that allows a more accurate construction of the sequence. This new recurrence relation allows us to decompose the recurrence matrix of the sequence as the product of three structured matrices. By computing the eigenvalues of the recurrence matrix, we obtain the zeros of the Gegenbauer–Sobolev polynomials. However, the eigenvalue problem formulated with this recurrence matrix is ill-conditioned and thus cannot lead to reliable approximations of the actual zeros. We deal with this ill-conditioning by reformulating the eigenvalue problem as a generalized eigenvalue problem, involving the three structured matrices that decompose the recurrence matrix, and applying a balancing technique to this generalized eigenvalue problem.
In this paper, we propose a conforming virtual element method to approximate the eigenfunctions and eigenvalues of the two-dimensional Oseen eigenvalue problem in its classic velocity-pressure formulation. We employ divergence-conforming virtual element spaces, originally developed for the Stokes equations, to approximate the velocity, and piecewise constant functions for the pressure. Under standard mesh assumptions, we derive a priori error estimates for the proposed method using the compact operator theory. In particular, we provide a rigorous tracking of the constants involved in the analysis, with the aim of having a detailed characterization of the convergence in norm between the solution operators. We report a set of numerical tests to confirm the theoretical results.
Abstract A classical problem in data-driven model order reduction (MOR) of linear time-invariant (LTI) systems is the preservation of structural properties of the underlying large-scale dynamics. When dealing with MOR based on transfer function measurements, one relevant problem is how to force the reduced-order model (ROM) to inherit the asymptotic stability of the reference system, i.e., to enforce the poles of the ROM transfer function to have strictly negative real part. In this work, we tackle the more general problem of placing such poles in arbitrary linear matrix inequality (LMI) regions of the complex plane, which include a rich class of convex sets symmetric with respect to the real axis. LTI systems with poles constrained to this kind of regions are called $$\mathcal {D}$$ D -stable. Combining well-established results from control theory and recent developments in asymptotically stable rational approximation algorithms, we show that the problem can be solved efficiently via standard convex optimization routines. Several numerical testbenches of engineering interest confirm the effectiveness of the proposed methodology in practical applications.
With the popularity of wearable devices, real-time electrocardiogram (ECG) monitoring is becoming increasingly important in heart health management. However, ECG signals acquired by these devices are highly susceptible to noises arising in signal acquisition. Diagnostic accuracy may often be lowered by these interfering noises. In this paper, by partitioning an ECG signal into heartbeat segments and establishing an ECG tensor, we propose a two-layer Kalman filter-based variational method (TLKFVM) for ECG signal denoising. In the first layer of the denoising process, we apply the l_1 -norm regularized variational model to remove noises within each heartbeat, which is solved by a Kalman smoother (KS) based on the intra-heartbeat state space model (SSM). In the second layer, by utilizing the similarity of heartbeat segments, we apply the l_2 -norm regularized variational model to further remove residue noises during the evolution period between consecutive heartbeats, which is solved by a Kalman filter (KF) based on the inter-heartbeat SSM. We call the first layer as the intra-heartbeat denoising and the second layer as the inter-heartbeat denoising. The experiments demonstrate the proposed method’s advantages in denoising quality, computational efficiency, and adaptivity when compared with the state-of-the-art methods for ECG denoising.
This study introduces a novel nonlinear parabolic equation characterized by a variable growth structure, specifically designed for image restoration and enhancement. Our approach builds upon classical models that employ variable exponent operators, extending their capabilities by incorporating a newly developed nonlinear operator featuring a double-phase flux with variable growth. This novel formulation allows for greater adaptability in modeling complex image structures while maintaining key elements like textures and corners. To establish a solid theoretical foundation, we begin by investigating the solvability of the proposed model. Utilizing the framework of variable exponent Lebesgue and Sobolev spaces, we develop an appropriate functional setting that enables rigorous mathematical analysis. The initial focus of our study is to ensure the well-posedness of the model. As a key result, we employ an approximation approach to prove the existence of a positive weak solution called solution obtained as limit of approximations (SOLA). Beyond theoretical considerations, we validate our model through practical applications in image processing. We conduct extensive numerical experiments on grayscale images to assess its performance in denoising and contrast enhancement. Additionally, we conducted evaluations on a collection of magnetic resonance imaging (MRI) scans, further demonstrating its applicability in medical imaging. Our experimental findings demonstrate that the proposed model consistently outperforms current state-of-the-art approaches, achieving enhanced computational efficiency and greater robustness while delivering superior results in both qualitative visual assessment and quantitative evaluation metrics. The findings emphasize the effectiveness of our model as a promising approach for tackling advanced image restoration and enhancement problems.
Legendre spectral differentiation is a basic tool in the numerical solution of differential equations. More precise information on spectral differentiation errors is important for deriving reliable error estimates for related numerical algorithms. However, existing studies primarily focus on asymptotic error estimates for polynomial approximations of specific singular functions and lack sharp pointwise error estimates for functions with interior or endpoint singularities in fractional spaces. In this work, we present explicit and sharp pointwise error estimates for Legendre spectral differentiation of functions with limited regularity in fractional spaces. We start by specifying a fractional-space setting suited to the pointwise error analysis in order to deal with the pointwise error estimates. We then derive explicit upper error bounds for Legendre spectral differentiation of functions with interior or endpoint singularities. Numerical experiments are provided to demonstrate the sharpness of our results.
The large-scale inverse eigenvalue problem arising from the fractional order control system presents mathematical and computational challenges due to its expensive computational costs and nonconvex nature. Considering that a smaller feedback control force implies less energy consumption and lower noise influence in control applications of the actual system, we transform the large-scale inverse eigenvalue problem into a matrix-based nonconvex optimization problem, which transcends the limitations of the vector-based optimization problem. To solve this nonconvex optimization problem, we propose an effective probability criterion and present a matrix-based randomized gradient descent method. The proposed method operates directly on the matrix form without vectorization and only needs to update a single row of the parameter matrix per iteration in practice, which makes it more suitable for the computation of large-scale nonconvex optimization problem. Then, we prove the smoothness of the objective function and further analyze the convergence of the proposed method. Finally, numerical experiments demonstrate the validity of our approach against traditional optimization-based methods and confirm the effectiveness of our approach in solving large-scale inverse eigenvalue problem arising from the fractional order control system.
This paper develops a new parameterized block (PB) preconditioner for solving the complex symmetric linear systems. We analyze the convergence behavior of the corresponding iteration scheme and derive the spectral properties of the preconditioned matrix. Numerical experiments are conducted to verify the effectiveness of the proposed method, demonstrating that the PB preconditioner significantly accelerates the convergence of the GMRES method when applied to complex symmetric linear systems.
The weighted and shifted backward differentiation formula (WSBDF) with a weighted parameter is an improved BDF-type method introduced by Akrivis et al. [IMA J. Numer. Anal., 45(6):3207-3234, 2025]. By introducing the weighted parameter β , it not only preserves the stability of the classical BDF method but also enhances the stability region and flexibility of the method. This paper presents a class of WSBDF schemes up to the fifth order for semilinear parabolic equations. Unlike the Nevanlinna-Odeh multiplier technique [Numer. Funct. Anal. Optim., 3:377-423, 1981] and explicit uniform multiplier technique [SIAM J. Numer. Anal., 62(4):1609-1637, 2024] based on Dahlquist’s G-stability theory [BIT, 18:384-401, 1978], we apply the discrete orthogonal convolution kernels technique to propose a straightforward discrete energy method and establish L^2 norm stability as well as error estimates for the high-order WSBDF schemes. Finally, numerical experiments are conducted to support our theoretical results.
We introduce the explicit strong stability preserving (SSP) two-derivative multistep Runge–Kutta (TDMSRK) methods. The order accuracy conditions and SSP theory for the TDMSRK methods are developed. By comparing the SSP coefficients of TDMSRK methods, Runge–Kutta schemes, two-derivative Runge–Kutta schemes, and general linear methods, it is indicated that the TDMSRK schemes have the largest effective SSP coefficient at the same order of accuracy. Some classical tests demonstrate the numerical stability of the TDMSRK methods on the Euler equation. Furthermore, the TDMSRK methods can achieve the expected order of accuracy and exhibit high computational efficiency in solving the Euler equation.
The Kaczmarz-type methods, also called as the algebraic reconstruction techniques, have been studied and applied due to their high computational efficiency and good convergence property for solving very large systems of linear equations. Recently, the residual-based extended Kaczmarz method was proposed by Bao et al. [9] to solve the inconsistent linear systems. In theoretical framework, based on the condition that the i_k-1 -th element of the residual vector r_k in the k-th iteration step is equal to zero, the authors obtained the convergence theorem. However, this element is not always equal to zero, making the basic hypothesis of the convergence theorem be not satisfied, so that the convergence conclusion does not hold true. According to this meticulous observation, we give a correct and detailed analysis for the convergence property of the residual-based extended Kaczmarz method. As a result, we provide a corrected convergence theory for this iteration method. Also, we examine these results by a numerical example.
We present a mixed weak Galerkin finite element method (WG-FEM) for singularly perturbed biharmonic problems on the unit square domain with clamped boundary conditions. By introducing an auxiliary variable, the fourth-order equation is reformulated as a coupled second-order system. A layer-adapted Shishkin mesh is employed to efficiently capture boundary layers. We derive parameter-uniform error bounds in both the energy and balanced norms, proving convergence independent of the perturbation parameter. Numerical results are provided to confirm the theoretical analysis and to demonstrate the effectiveness of the mixed WG-FEM with layer-adapted meshes.
We study a symmetric Q_1 -finite volume element scheme for anisotropic diffusion problems on unstructured convex quadrilateral meshes. This scheme was originally suggested in [39] where the diffusion coefficient is a scalar and the analysis was done for Poisson equations on a uniform rectangular mesh. In this paper, we prove the symmetry and positive definiteness of the global bilinear form, by which the well-posedness of the scheme is established on an arbitrary convex quadrilateral mesh with any mesh size and a full anisotropic diffusion tensor. Under a weak mesh assumption, which covers the structured convex quadrilateral mesh, we present a simply analysis (different from the standard cell analysis approach) for the proof of L^2 coercivity of the symmetric scheme. Further, by Taylor expansion, the optimal L^2 and H^1 error estimates are verified on the bisection quadrilateral mesh with a full anisotropic diffusion tensor. Consequently, we improve the theoretical results in [39]. Finally, several numerical examples are provided to validate the theoretical findings.
We propose a new deterministic block Kaczmarz-type method for solving large-scale consistent systems of linear equations. In the proposed method, the block of the subsystem of linear equations is determined by an easier to compute method, and it is pseudoinverse-free to solve the subsystem. To further accelerate the convergence rate of the proposed method, we present an accelerated version of this method using momentum. Linear convergence of the proposed method and its acceleration are proved. Numerical results show that the proposed method and its acceleration are effective and very fast for the test problems.
